10 Polynomials and numerical semigroups
  
  
  10.1 Generating functions or Hilbert series
  
  Let  S  be  a numerical semigroup. The Hilbert series or generating function
  associated  to S is H_S(x)=∑_s∈ Sx^s (actually it is the Hilbert function of
  the ring K[S] with K a field). See for instance [Mor14].
  
  10.1-1 NumericalSemigroupPolynomial
  
  NumericalSemigroupPolynomial( s, x )  function
  
  s  is  a  numerical semigroups and x a variable (or a value to evaluate in).
  The output is the polynomial 1+(x-1)∑_s∈ N∖ S x^s, which equals (1-x)H_S(x).
  
    Example  
    gap> x:=X(Rationals,"x");;
    gap> s:=NumericalSemigroup(5,7,9);;
    gap> NumericalSemigroupPolynomial(s,x);
    x^14-x^13+x^12-x^11+x^9-x^8+x^7-x^6+x^5-x+1
  
  
  10.1-2 IsNumericalSemigroupPolynomial
  
  IsNumericalSemigroupPolynomial( f )  function
  
  f  is  a  polynomial  in  one variable. The output is true if there exists a
  numerical  semigroup  S  such  that  f  equals  (1-x)H_S(x),  that  is,  the
  polynomial associated to S (false otherwise).
  
    Example  
    gap> x:=X(Rationals,"x");;
    gap> s:=NumericalSemigroup(5,6,7,8);;
    gap> f:=NumericalSemigroupPolynomial(s,x);
    x^10-x^9+x^5-x+1
    gap> IsNumericalSemigroupPolynomial(f);
    true
  
  
  10.1-3 NumericalSemigroupFromNumericalSemigroupPolynomial
  
  NumericalSemigroupFromNumericalSemigroupPolynomial( f )  function
  
  f  is  a  polynomial  associated  to a numerical semigroup (otherwise yields
  error).  The  output  is  the  numerical  semigroup  S  such  that  f equals
  (1-x)H_S(x).
  
    Example  
    gap> x:=X(Rationals,"x");;
    gap> s:=NumericalSemigroup(5,6,7,8);;
    gap> f:=NumericalSemigroupPolynomial(s,x);
    x^10-x^9+x^5-x+1
    gap> NumericalSemigroupFromNumericalSemigroupPolynomial(f)=s;
    true
  
  
  10.1-4 HilbertSeriesOfNumericalSemigroup
  
  HilbertSeriesOfNumericalSemigroup( s, x )  function
  
  s is a numerical semigroup and x a variable (or a value to evaluate in). The
  output is the series ∑_s∈ S x^s. The series is given as a rational function.
  
    Example  
    gap> x:=X(Rationals,"x");;
    gap> s:=NumericalSemigroup(5,7,9);;
    gap> HilbertSeriesOfNumericalSemigroup(s,x);
    (x^14-x^13+x^12-x^11+x^9-x^8+x^7-x^6+x^5-x+1)/(-x+1)
  
  
  10.1-5 GraeffePolynomial
  
  GraeffePolynomial( p )  function
  
  p is a polynomial. Computes the Graeffe polynomial of p. Needed to test if p
  is a cyclotomic polynomial (see [BD89]).
  
    Example  
    gap> x:=Indeterminate(Rationals,1);; SetName(x,"x");
    gap> GraeffePolynomial(x^2-1);
    x^2-2*x+1
  
  
  10.1-6 IsCyclotomicPolynomial
  
  IsCyclotomicPolynomial( p )  function
  
  p  is  a  polynomial.  Detects  if  p  is  a cyclotomic polynomial using the
  procedure given in [BD89].
  
    Example  
    gap> CyclotomicPolynomial(Rationals,3);
    x^2+x+1
    gap> IsCyclotomicPolynomial(last);
    true
  
  
  10.1-7 IsKroneckerPolynomial
  
  IsKroneckerPolynomial( p )  function
  
  p  is a polynomial. Detects if p is a Kronecker polynomial, that is, a monic
  polynomial  with  integer  coefficients  having  all  its  roots in the unit
  circumference,  or  equivalently,  a  product of cyclotomic polynomials. The
  current  implementation  has  been  done  with A. Herrera-Poyatos, following
  [BD89].
  
    Example  
    gap> x:=X(Rationals,"x");;
    gap>  s:=NumericalSemigroup(3,5,7);;
    gap>  t:=NumericalSemigroup(4,6,9);;
    gap> p:=NumericalSemigroupPolynomial(s,x);
    x^5-x^4+x^3-x+1
    gap> q:=NumericalSemigroupPolynomial(t,x);
    x^12-x^11+x^8-x^7+x^6-x^5+x^4-x+1
    gap> IsKroneckerPolynomial(p);
    false
    gap> IsKroneckerPolynomial(q);
    true
  
  
  10.1-8 IsCyclotomicNumericalSemigroup
  
  IsCyclotomicNumericalSemigroup( s )  function
  
  s  is  a numerical semigroup. Detects if the polynomial associated to s is a
  Kronecker polynomial.
  
    Example  
    gap> l:=CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber(21);;
    gap> ForAll(l,IsCyclotomicNumericalSemigroup);
    true
  
  
  10.1-9 IsSelfReciprocalUnivariatePolynomial
  
  IsSelfReciprocalUnivariatePolynomial( p )  function
  
  p  is  a  univariate polynomial. Detects if p is selfreciprocal. A numerical
  semigroup  is  symmetric  if  and only if it is selfreciprocal, [Mor14]. The
  current implementation is due to A. Herrera-Poyatos.
  
    Example  
    gap> l:=IrreducibleNumericalSemigroupsWithFrobeniusNumber(13);;
    gap> x:=X(Rationals,"x");;
    gap> ForAll(l, s->
    > IsSelfReciprocalUnivariatePolynomial(NumericalSemigroupPolynomial(s,x)));
    true
  
  
  
  10.2 Semigroup of values of algebraic curves
  
  Let  f(x,y)∈  K[x,y], with K an algebraically closed field of characteristic
  zero.  Let  f(x,y)=y^n+a_1(x)y^n-1+dots+a_n(x)  be  a  nonzero polynomial of
  K[x][y].  After  possibly  a  change  of variables, we may assume that, that
  deg_x(a_i(x))le  i-1  for  all  i∈{1,...,  n}.  For  g∈ K[x,y] that is not a
  multiple  of  f, define mathrmint(f,g)=dim_ K frac K[x,y](f,g). If f has one
  place  at  infinity,  then the set {mathrmint(f,g)∣ g∈ K[x,y]∖(f)} is a free
  numerical semigroup (and thus a complete intersection).
  
  10.2-1 SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity
  
  SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity( f )  function
  
  f  is  a  polynomial  in  the  variables  X(Rationals,1) and X(Rationals,2).
  Computes    the    semigroup    {mathrmint(f,g)∣   g∈   K[x,y]∖(f)},   where
  mathrmint(f,g)=dim_  K  (  K[x,y]/(f,g)).  The algorithm checks if f has one
  place  at infinity. If the extra argument "all" is given, then the output is
  the  δ-sequence  and  approximate  roots  of  f.  The method is explained in
  [AGS16].
  
    Example  
    gap> x:=Indeterminate(Rationals,1);; SetName(x,"x");
    gap> y:=Indeterminate(Rationals,2);; SetName(y,"y");
    gap> f:=((y^3-x^2)^2-x*y^2)^4-(y^3-x^2);;
    gap> SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity(f,"all");
    [ [ 24, 16, 28, 7 ], [ y, y^3-x^2, y^6-2*x^2*y^3+x^4-x*y^2 ] ]
  
  
  10.2-2 IsDeltaSequence
  
  IsDeltaSequence( l )  function
  
  l  is  a  list  of positive integers. Assume that l equals a_0,a_1,dots,a_h.
  Then l is a δ-sequence if gcd(a_0,..., a_h)=1, ⟨ a_0,⋯, a_s⟩ is free, a_kD_k
  >  a_k+1D_k+1  and  a_0>  a_1  >  D_2  >  D_3  > ... > D_h+1, where D_1=a_0,
  D_k=gcd(D_k-1,a_k-1).
  
  Every  δ-sequence  generates  a numerical semigroup that is the semigroup of
  values of a plane curve with one place at infinity.
  
    Example  
    gap> IsDeltaSequence([24,16,28,7]);
    true
  
  
  10.2-3 DeltaSequencesWithFrobeniusNumber
  
  DeltaSequencesWithFrobeniusNumber( f )  function
  
  f  is  a  positive  integer.  Computes the set of all δ-sequences generating
  numerical semigroups with Frobenius number f.
  
    Example  
    gap> DeltaSequencesWithFrobeniusNumber(21);
    [ [ 8, 6, 11 ], [ 10, 4, 15 ], [ 12, 8, 6, 11 ], [ 14, 4, 11 ],
      [ 15, 10, 4 ], [ 23, 2 ] ]
  
  
  10.2-4 CurveAssociatedToDeltaSequence
  
  CurveAssociatedToDeltaSequence( l )  function
  
  l  is  a  δ-sequence.  Computes  a curve in the variables X(Rationals,1) and
  X(Rationals,2) whose semigroup of values is generated by the l.
  
    Example  
    gap> CurveAssociatedToDeltaSequence([24,16,28,7]);
    y^24-8*x^2*y^21+28*x^4*y^18-56*x^6*y^15-4*x*y^20+70*x^8*y^12+24*x^3*y^17-56*x^\
    10*y^9-60*x^5*y^14+28*x^12*y^6+80*x^7*y^11+6*x^2*y^16-8*x^14*y^3-60*x^9*y^8-24\
    *x^4*y^13+x^16+24*x^11*y^5+36*x^6*y^10-4*x^13*y^2-24*x^8*y^7-4*x^3*y^12+6*x^10\
    *y^4+8*x^5*y^9-4*x^7*y^6+x^4*y^8-y^3+x^2
    gap> SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity(last,"all");
    [ [ 24, 16, 28, 7 ], [ y, y^3-x^2, y^6-2*x^2*y^3+x^4-x*y^2 ] ]
  
  
  10.2-5 SemigroupOfValuesOfPlaneCurve
  
  SemigroupOfValuesOfPlaneCurve( f )  function
  
  f  is  a  polynomial in the variables X(Rationals,1) and X(Rationals,2). The
  singular  package  is  mandatory.  Either  by  loading it prior to numerical
  semigroups  or  by using NumSgpsUseSingular(). If f is irreducible, computes
  the semigroup {mathrmint(f,g)∣ g∈ K[x,y]∖(f)}, where mathrmint(f,g)=dim_ K (
  K[[x,y]]/(f,g)). If it has two components, the output is the value semigroup
  in  two  variables, and thus a good semigroup. If there are more components,
  then the output is that of semigroup in the alexpoly singular library.
  
    Example  
    gap> x:=X(Rationals,"x");; y:=X(Rationals,"y");;
    gap> f:= y^4-2*x^3*y^2-4*x^5*y+x^6-x^7;
    -x^7+x^6-4*x^5*y-2*x^3*y^2+y^4
    gap> SemigroupOfValuesOfPlaneCurve(f);
    <Numerical semigroup with 3 generators>
    gap> MinimalGenerators(last);
    [ 4, 6, 13 ]
    gap> f:=(y^4-2*x^3*y^2-4*x^5*y+x^6-x^7)*(y^2-x^3);;
    gap> SemigroupOfValuesOfPlaneCurve(f);
    <Good semigroup>
    gap> MinimalGenerators(last);
    [ [ 4, 2 ], [ 6, 3 ], [ 13, 15 ], [ 29, 13 ] ]
  
  
  10.2-6 SemigroupOfValuesOfCurve_Local
  
  SemigroupOfValuesOfCurve_Local( arg )  function
  
  The  function admits one or two parameters. In any case, the first is a list
  of  polynomials pols. And the second can be the string "basis" or an integer
  val.
  
  If  only  one argument is given, the output is the semigroup of all possible
  orders of K[[pols]] provided that K[[x]]/K[[pols]] has finite length. If the
  second  argument  "basis"  is given, then the output is a (reduced) basis of
  the  algebra  K[[pols]]  such that the orders of the basis elements generate
  minimally  the  semigroup  of  orders of K[[pols]]. If an integer val is the
  second argument, then the output is a polynomial in K[[pols]] with order val
  (fail  if  there is no such polynomial, that is, val is not in the semigroup
  of values).
  
  The method is explained in [AGSM17].
  
    Example  
    gap> x:=Indeterminate(Rationals,"x");;
    gap> SemigroupOfValuesOfCurve_Local([x^4,x^6+x^7,x^13]);
    <Numerical semigroup with 4 generators>
    gap> MinimalGeneratingSystem(last);
    [ 4, 6, 13, 15 ]
    gap> SemigroupOfValuesOfCurve_Local([x^4,x^6+x^7,x^13], "basis");
    [ x^4, x^7+x^6, x^13, x^15 ]
    gap> SemigroupOfValuesOfCurve_Local([x^4,x^6+x^7,x^13], 20);
    x^20
  
  
  10.2-7 SemigroupOfValuesOfCurve_Global
  
  SemigroupOfValuesOfCurve_Global( arg )  function
  
  The  function admits one or two parameters. In any case, the first is a list
  of  polynomials pols. And the second can be the string "basis" or an integer
  val.
  
  If  only  one argument is given, the output is the semigroup of all possible
  degrees  of  K[pols]  provided  that  K[x]/K[pols] has finite length. If the
  second  argument  "basis"  is given, then the output is a (reduced) basis of
  the  algebra  K[pols]  such  that the degrees of the basis elements generate
  minimally  the  semigroup  of  degrees  of K[pols]. If an integer val is the
  second  argument, then the output is a polynomial in K[pols] with degree val
  (fail  if  there is no such polynomial, that is, val is not in the semigroup
  of values).
  
  The method is explained in [AGSM17].
  
    Example  
    gap> x:=Indeterminate(Rationals,"x");;
    gap> SemigroupOfValuesOfCurve_Global([x^4,x^6+x^7,x^13]);
    <Numerical semigroup with 3 generators>
    gap> MinimalGeneratingSystem(last);
    [ 4, 7, 13 ]
    gap> SemigroupOfValuesOfCurve_Global([x^4,x^6+x^7,x^13],"basis");
    [ x^4, x^7+x^6, x^13 ]
    gap> SemigroupOfValuesOfCurve_Global([x^4,x^6+x^7,x^13],12);
    x^12
    gap> SemigroupOfValuesOfCurve_Global([x^4,x^6+x^7,x^13],6);
    fail
  
  
  10.2-8 GeneratorsModule_Global
  
  GeneratorsModule_Global( A, M )  function
  
  A and M are lists of polynomials in the same variable. The output is a basis
  of  the  ideal M K[A], that is, a set F such that deg(F) generates the ideal
  deg(M  K[A])  of  deg(K[A]),  where  deg  stands  for  degree. The method is
  explained in [AAGS17].
  
    Example  
    gap> t:=Indeterminate(Rationals,"t");;
    gap> A:=[t^6+t,t^4];;
    gap> M:=[t^3,t^4];;
    gap> GeneratorsModule_Global(A,M);
    [ t^3, t^4, t^5, t^6 ]
  
  
  10.2-9 GeneratorsKahlerDifferentials
  
  GeneratorsKahlerDifferentials( A, M )  function
  
  A   is   a  list  of  polynomials  in  the  same  variable.  The  output  is
  GeneratorsModule_Global(A,M),  with M the set of derivatives of the elements
  in A.
  
    Example  
    gap> t:=Indeterminate(Rationals,"t");;
    gap> GeneratorsKahlerDifferentials([t^3,t^4]);
    [ t^2, t^3 ]
  
  
  10.2-10 IsMonomialNumericalSemigroup
  
  IsMonomialNumericalSemigroup( S )  property
  
  S is a numerical semigroup. Tests whether S a monomial numerical semigroup.
  
  Let  R  a  Noetherian  ring  such  that  K  ⊆  R  ⊆  K[[t]], K is a field of
  characteristic zero, the algebraic closure of R is K[[t]], and the conductor
  (R  :  K[[t]])  is  not zero. If v : K((t))-> Z is the natural valuation for
  K((t)), then v(R) is a numerical semigroup.
  
  Let  S  be  a  numerical semigroup minimally generated by {n_1,...,n_e}. The
  semigroup  ring  associated  to  S is K[[S]]=K[[t^n_1,...,t^n_e]]. A ring is
  called  a  semigroup  ring  if  it is of the form K[[S]], for some numerical
  semigroup S. We say that S is a monomial numerical semigroup if for any R as
  above with v(R)=S, R is a semigroup ring. See [Mic02] for details.
  
    Example  
    gap> IsMonomialNumericalSemigroup(NumericalSemigroup(4,6,7));
    true
    gap> IsMonomialNumericalSemigroup(NumericalSemigroup(4,6,11));
    false